Characterizations of some wreath products of groups by their endomorphism semigroups

Let A be a cyclic group of order p(n), where p is a prime, and B be a finite abelian group or a finite p-group which is determined by its endomorphism semigroup in the class of all groups. It is proved that under these assumptions the wreath product A Wr B is determined by its endomorphism semigroup in the class of all groups. It is deduced from this result that if A, B, A(0),..., A(n) are finite abelian groups and A(0),..., An are p-groups, p prime, then the wreath products A Wr B and A(n) Wr (...( Wr ( A(1) Wr A(0)))...) are determined by their endomorphism semigroups in the class of all groups.